372 Brazilian Journal of Physics, vol. 36, no. 2A, June, 2006
Electron-Phonon Scattering in Graded Quantum Dots
G. S. Diniz, Fanyao Qu, O. O. Diniz Neto, and A. M. Alcalde Instituto de F´ısica, Universidade Federal de Uberlˆandia,
Caixa Postal 593, 38400-902, Uberlˆandia-MG, Brazil
Received on 4 April, 2005
Theoretical calculations of electron-phonon scattering rates in GaAs/AlxGa1−xAs spherical quantum dots have been performed by means of effective mass approximation in the frame of finite element method. The in-fluence of a roughness interface and external magnetic fields are analysed for different scattering rate transition. Our results open interesting channels for electron dephasing times manipulation.
Keywords: Theoretical calculations; Electron-phonon scattering; GaAs/AlxGa1−xAs
I. INTRODUCTION
One of the most common procedures of fabricating semi-conductor low-dimensional structures is by epitaxial grow-ing of compositionally graded alloys as AlxGa1−xAs. Thus,
structures that confine electrons are made by changing the aluminumxfraction during crystal growth leading to a com-positional graded alloy. The resulting band structure varia-tion produces a spatially varying conducvaria-tion band minimum. Hence, an electron added to the conduction band through dop-ing, optical excitation or electrical injection, “sees” a position-dependent potential. By varying the concentration appropri-ately, one can engineer confining potentials that restrict elec-tron motion in a particular spatial direction. Intentional grad-ing can be used to control the strain in quantum dot struc-tures [1], reduce the electron capture times [2] or modify the electron-phonon scattering [3,8].
In this work, theoretical calculations of electron-phonon scattering rates in GaAs/AlxGa1−xAs spherical quantum dots
(SQD’s) have been performed by means of an effective mass approximation in the frame of the finite element method (FEM)[4]. The influence of a symmetry breaking of the wave function on the electron dephasing times due to the electron-phonon interaction is investigated for various SQD sizes and grading profiles. It is demonstrated that the electron-acoustic phonon scattering rates strongly depends upon both SQD size and magnetic field strength. For different allowed transitions, given by the selection rules∆m= 0 and∆m= 1, wheremis the magnetic quantum number of the conduction electron states, the scattering rates present opposite magnetic field dependent behaviour, which can be tuned by a properly choice of mag-netic field strength and compositional grading.
II. THEORY AND RESULTS
The Hamiltonian for a single conduction electron in pres-ence of magnetic fieldB= (0,0,B), with a vector potentialA =(-y, x,0)B/2, is H=H0+HB+V, whereH0includes the kinetic energy. The termHBis given by-eBmh/2m∗+e2B2
(x2+y2)/8m∗, withm=0,±1,±2..., andm* is the electron effective mass. The confinement graded potentialV [5] is de-scribed by the following equations (ineVunits):
V(ρ) =
0 ρ≤R−ξ,
0.6(1.155y+0.37y2) R−ξ<ρ≤R+ξ, 0.6(1.155x+0.37x2) ρ>R+ξ,
whereR represents the quantum dot radius, ξ the interface thickness, x is the aluminum concentration (in this work, x=0.3) andy=x(ρ+ξ-R)/R, representing the linear mo-lar fraction in the interface region.
The electron effective mass is described bym∗/me=0.067
for GaAs, m∗/me=0.067+0.083y for the interface region,
m∗/me=0.067+0.083xfor AlxGa1−xAs barrier. At low
tem-peratures, carrier relaxation rates are determined mainly by intraband transitions, involving acoustic phonon emission.
The electron-phonon scattering rateWfor electron transi-tions from an initial (n,m)to a final (n′,m′)state with emission of acoustic phonon of energyvq is calculated by the Fermi Golden Rule [6]:
Γ=2π
~
Z
d3q|M(q)|2δ(En,m−En′,m′−~vq). (1)
Here q is the phonon wave vector, v is the longitudinal sound speed, andM (q)is the matrix element for electronic scattering accompanied by the emission of a phonon
M(q) =iπC(q)[(qx+iqy)δ∆m=1G1+2qzδ∆m=0G2], (2) where the overlap integrals are
G1=
Z
ρdρdzRn′,m′(ρ,z)ρRn,m(ρ,z), (3)
and
G2=
Z
ρdρdzRn′,m′(ρ,z)zRn,m(ρ,z). (4)
In the above equations∆m=m− −m’.C(q)is the coupling constant for piezoelectric (PE) and deformation potential (DP) mechanisms [7], andRn,mare the orbital wave functions,
G. S. Diniz et al. 373
FIG. 1: Electron-acoustic phonon scattering rates as a function of SQD radius for∆m= 0 (solid line) and∆m= 1 (dashed line): (a) DP coupling and (b) PE coupling.
Figures 1(a) and 1(b) show the electron-acoustic phonon scattering ratesW as a function of the quantum dot radiusR for DP and PE mechanisms, respectively. In both situations we considerB = 0 T. The data in Figs. 1(a) and (b) are for transitions from the first excited state (n = 1, m= 1) to the ground state (n= 0,m= 0)∆m= 1 (dashed line) and for the transition (n= 1, m = 0) to (n= 0,m= 0)∆m= 0 (solid line), according to the selection rules given in Eq. (2). In analysing the data in Fig. 1, some interesting aspects can be observed: as the quantum dot radiusRincrease the rateW decreases for both DP and PE mechanisms. This behaviour is due to the decreasing of electron confinement in the SQD that reduces the value of the overlap integrals and consequently decreases the magnitude of the scattering ratesW.
In Fig. 2 (a) and 2 (b) is shown the interface thickness (ξ) influence on the transition rates, for∆m= 1 (dashed line) and
∆m= 0 (solid line) transitions forB= 0 T. We have chosen SQDs ofR= 120 ˚A andξvarying from 0-24 ˚A. We can re-alize that for both DP and PE mechanisms, when the inter-face thickness is increased the electron “feels” a strong spatial confinement leading to a “blue shift” that lift the value of the overlap integrals, increasing the scattering ratesW.
When is analysed the influence of an external magnetic field upon the SQD, we can see from Fig. 3 that the scattering rates present a small dependency for∆m= 0, since electronic lev-els with ∆m= 0 are weakly perturbed by the applied mag-netic field. In opposite, the behaviour of∆m = 1 transition
FIG. 2: Electron-acoustic phonon scattering rates as a function of interface thickness forR= 120 ˚A and∆m= 0 (solid line) and∆m= 1 (dashed line): (a) DP coupling and (b) PE coupling.
374 Brazilian Journal of Physics, vol. 36, no. 2A, June, 2006
is strongly modified by external magnetic field, this is due to the extra electron confinement, it is clear that the combination of magnetic and spatial confinements will determine the value of the overlap. In general, when the spatial confinement is strong, the magnetic field produces an additional spatial con-finement of the electronic wave functions which is more sensi-ble for∆m= 1 transition. This behaviour, it is clearly showed in Figs. 3 (a) and (b) for both DP and PE mechanisms for a SQD ofR= 120 ˚A,ξ= 18 ˚A. Solid line and dashed line rep-resenting respectively the∆m= 0 and∆m= 1 transitions.
III. CONCLUSION
In conclusion, the effects of a graded interface, SQDs radius and magnetic field upon electron-phonon scattering rates of
AlxGa1−xAs/GaAs SQDs have been investigated in the frame
of FEM. It was found that a roughness interface can strongly modify the rates for both DP and EP mechanism. Further the magnetic field and SQD radius play an important rule for a given∆m= 0, 1 transitions. Indeed, our findings open up new insights for a more precise tuning and control of the carrier de-phasing time in semiconductor quantum dots with an impact on device designs.
Acknowledgement
This work has been supported by Fundac¸˜ao de Amparo `a Pesquisa do Estado de Minas Gerais (FAPEMIG) and Con-selho Nacional de Pesquisa (CNPq).
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[6] P. A. Knipp and T. L. Reinecke, Phys. Rev. B,52, 5993 (1995). [7] The coupling constantsC(q)are defined by
CDPq =−i
s ~
2ρmVωqeq·qΞ,
and
CqPE=
s ~
2ρmVωqe EP
ε ,
whereωq=vq,vis the sound velocity,qis the phonon wave vec-tor.Ξis the deformation constant,EP is the piezoelectric con-stant,εis the dielectric constant, andρMis the mass density of material. The constants as sound speed, mass density, dielectric constant, and effective mass vary linearly through the interface region. See Wenhuiet al, Phys. Rev. B,49, 14403 (1994) and references therein.
